Let G be a linearly reductive group over a field k, and let R be a kalgebra with a rational action of G. Given rational R-G-modules M and N, we define for the induced G-action on HomR(M,N) a generalized Reynolds operator, which exists even if the action on HomR(M,N) is not rational. Given an R-module homomorphism M → N, it produces, in a naturalway, an R-module homomorphism which is G-equivariant. We use this generalized Reynolds operator to study properties of rational R-G modules. In particular, we prove that if M is invariantly generated (i.e. M = R·MG), then MG is a projective (resp. flat) RG-module provided that M is a projective (resp. flat) R-module. We also give a criterion whether an R-projective (or R-flat) rational R-G-module is extended from an RG-module.
Proceedings of the American Mathematical Society 133(10), pp.2865-2871